Rarefaction-singular shock dynamics for conserved volume gravity driven particle-laden thin film

Wang, L.; Mavromoustaki, Aliki; Bertozzi, Andrea L.; Urdaneta, Gilberto; Huang, Kexuan

doi:10.1063/1.4913851

Cited by 7 publications

(15 citation statements)

References 22 publications

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“…In section 5 we derive theoretical results for the long-time behavior of monodisperse suspensions in the settled regime, where the concentration of particles uniformly approaches a critical concentration independent of initial conditions. The asymptotic behavior of the front positions is derived, extending the existing results for the high-concentration [19,20] and dilute limit [12]. The results are then extended qualitatively to the bidisperse problem.…”

Section: Introductionmentioning

confidence: 69%

“…The theory for the dilute limit of the dynamic model was studied in [12], where an exact solution for the rarefaction can be found, and the particle front was shown to evolve as x p = (C p t) 1/3 + T 0 + O(t −1/3 ) for some constant T 0 . A similar behavior occurs in the high-concentration limit, where one obtains a rarefaction and a singular shock [19] and the exponent α in x f (t) ∼ t α is perturbed slightly from 1/3 due to the accumulation of mass at the front.…”

Section: Long-term Behavior: Monodispersementioning

confidence: 81%

“…Previous investigations have explored the transition between settled and ridged regimes [6]. The experimental data considered [10] has been collected for angles and concentrations within the settled regime (the rarefaction-singular shock pairs expected in the ridged regime [19] have not been fully explored in experiments). The data tracked is the position of the three fronts obtained from imaging, as the height and concentration profiles are difficult to determine accurately.…”

Section: Constant Volume Solutionsmentioning

confidence: 99%

“…The key observation, used in [19] to approximate the rarefaction in the ridged regime, is that the concentration within the first rarefaction approaches the critical concentration φ c as t → ∞. If we define φ(ξ) = n(ξ)/h(ξ), then it can be shown [19] that necessarily φ(0) = lim ξ→0 n(ξ)/h(ξ) = φ c if n(0) = h(0) = 0. Since x p (t)/t → 0 as t → ∞, it must be that φ → φ c uniformly for x ∈ [0, x p (t)] as t → ∞.…”

Section: Long-term Behavior: Monodispersementioning

confidence: 99%

“…The Riemann problem has been studied for the monodisperse model [9,20], showing the existence of double shock solutions in the settled regime and a transition to singular shocks that occurs in the ridged regime, where particles accumulate at the front. The corresponding rarefaction-singular shock solutions that arise for constant volume initial conditions have also been studied [19].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A conservation law model for bidensity suspensions on an incline

Wong

Bertozzi

2016

Physica D: Nonlinear Phenomena

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We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravity-induced settling and shear induced migration. We propose an extension a recent model [N. Murisic, B. Pausader, D. Peschka, and A. L. Bertozzi, Dynamics of particle settling and resuspension in viscous liquid films, Journal of Fluid Mechanics 717 (2013), 203-231] for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits three-shock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the single-species case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefaction-shock pairs, and present a comparison to recent experimental results. The long-time behavior of solutions is identified for settled mono-and bidisperse suspensions and some leading-order asymptotics are derived in the single-species case for moderate concentrations.

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Section: Introductionmentioning

confidence: 69%

Section: Long-term Behavior: Monodispersementioning

confidence: 81%

Section: Constant Volume Solutionsmentioning

confidence: 99%

Section: Long-term Behavior: Monodispersementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A conservation law model for bidensity suspensions on an incline

Wong

Bertozzi

2016

Physica D: Nonlinear Phenomena

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A mathematical treatment of the bump structure of particle-laden flows with particle features

Matsue

Tomoeda

2022

Japan J. Indust. Appl. Math.

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In this study, we consider particle-laden flows on an inclined plane under the effect of gravity. Previous experimental works have noted that a particle-rich ridge is generated near the contact line. We investigate the bump structure observed in particle-rich ridges in terms of Lax’s shock waves in the mathematical theory of conservation laws, explicitly considering the effect of particles with nontrivial radii on morphology of particle-laden flows. We also extract the dependence of radius and concentration of particles on the bump structure.

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Monodisperse particle-laden exchange flows in a vertical duct

Mirzaeian

Alba

2018

J. Fluid Mech.

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We study buoyancy-driven exchange flow of two mixtures in a vertical narrow duct (two-dimensional channel as well as pipe) theoretically. While the light mixture is assumed always to be a pure fluid, the heavy mixture can be selected as either a pure or a particle-laden fluid. A small width-to-length ratio considered for the duct ($\unicode[STIX]{x1D6FF}\ll 1$) has been used as the perturbation parameter in developing a lubrication model (negligible inertia). In particular, we have adopted the methodology of Zhou et al. (Phys. Rev. Lett., vol. 94, 2005, 117803) for free-surface particle-laden film flows and extended it to a lock exchange system in confined geometry under the Boussinesq approximation. The resulting model is in the form of the classical Riemann problem and has been solved numerically using a robust total variation diminishing finite difference scheme. Both pure and particle-laden cases are investigated in detail. It is observed that the interface between the two fluids takes a self-similar shape at long times. In the case that both heavy and light fluids are pure, the dynamics of the flow is governed by two dimensionless quantities, namely the Reynolds number, $Re$, and the viscosity ratio, $\unicode[STIX]{x1D705}$, of the light and heavy fluids. The interpenetration of the heavy and light layers increases with $Re$ but decreases with $\unicode[STIX]{x1D705}$. Also, the heights of the heavy and light fronts change with $\unicode[STIX]{x1D705}$ but remain unchanged with $Re$. In the case of the particle-laden flow, however, four additional dimensionless parameters emerge, namely the initial volume fraction of particles, $\unicode[STIX]{x1D719}_{0}$, the ratio of particle diameter to duct width, $r_{p}$, and the density ratios of particles to carrying fluid, $\unicode[STIX]{x1D709}$, and of light fluid to carrying fluid, $\unicode[STIX]{x1D702}$. The effect of these parameters on the dynamics of the flow has been quantified through a systematic approach. In the presence of solid particles, the interface between the heavy and light layers becomes more curved compared to the case of pure fluids. This modification occurs due to the change of heavy mixture viscosity alongside the duct. Novel particle-rich zones are further discovered in the vicinity of the advancing heavy and light fronts. These zones are associated with different transport rates of the fluid and solid particles. The degree of particle enrichment remains the same with $Re$, is enhanced by $\unicode[STIX]{x1D705}$, $r_{p}$ and $\unicode[STIX]{x1D702}$, and is slightly diminished with $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D709}$. On the other hand, the stretched exchange zone between the heavy and light fronts grows with $r_{p}$, $\unicode[STIX]{x1D702}$ and $Re$, but decays with $\unicode[STIX]{x1D719}_{0}$, $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D709}$.

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Rarefaction-singular shock dynamics for conserved volume gravity driven particle-laden thin film

Cited by 7 publications

References 22 publications

A conservation law model for bidensity suspensions on an incline

A conservation law model for bidensity suspensions on an incline

A mathematical treatment of the bump structure of particle-laden flows with particle features

Monodisperse particle-laden exchange flows in a vertical duct

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